XX.
GASEOUS ELECTRONICS
Academic and Research Staff
Prof. S. C. Brown
Prof. W. P. Allis
Prof. G. Bekefi
Prof. J. C. Ingraham
E. M. Mattison
J. J. McCarthy
W. J. Mulligan
Graduate Students
W. B. Davis
G. A. Garosi
L. D. Pleasance
T. T. Wilheit, Jr.
B. L. Wright
RESEARCH OBJECTIVES
A great deal of effort continues to be spent on the fundamental interaction of the particles of plasma physics. In this group we have concentrated on measuring diffusion
coefficients, collision probabilities, particularly in partially ionized and fully ionized
gases, and on the mechanism of production of radiation. Within the last few years we
have studied the production of microwave radiation, and we are pushing this toward
shorter and shorter wavelengths in the direction of the very long infrared. Our objective in this area is to study as many of the mechanisms involved in electron-atom and
electron-molecule collisions as are susceptible to our microwave, infrared, and probe
techniques.
S. C. Brown
A.
ELECTRON-ELECTRON
RELAXATION RATES AS DETERMINED FROM THE
OBSERVED TIME-DEPENDENT ELECTRON VELOCITY DISTRIBUTION
Experimental observations of the time-dependent electron velocity distribution function, f(v, t), during the early afterglow of an argon discharge have been previously
reported.1
Solution of an appropriate theoretical model showed good agreement with
both the observed steady-state distribution during the DC voltage pulse and the instantaneous energy loss rate at the beginning of the afterglow.
experiment
(argon pressure
-1
Torr
and electron
2
Under the conditions of the
density ~10
10
cm
-1
),
electron-
electron and electron-atom collisions are found to represent competing mechanisms
with comparable characteristic rates.
In interpreting the observed time dependence of
f(v, t), it
is desirable to examine quantities which to some extent separate these two
effects.
Thus, since mean electron energy is conserved by electron-electron inter-
actions, the rate of energy loss previously analyzed is largely determined by electronatom collisions.
Electron-electron effects,
however,
play an indirect role
in
the
cooling by replenishing f(v) at higher velocities where (in argon) the energy loss rate is
greater.
To obtain a more direct measure of the rate of electron-electron interaction, we
This work was supported by the Joint Services Electronics Programs (U. S. Army,
U. S. Navy, and U. S. Air Force) under Contract DA 36-039-AMC-03200(E).
QPR No. 84
137
(XX.
GASEOUS ELECTRONICS)
note that the ultimate effect of this interaction is the attainment after a few micro0
seconds of a nearly Maxwellian velocity distribution with a temperature (-10, 000 K)
well above that of the neutral gas. Parameters which in some sense measure the departure of f(v) from a Maxwellian should consequently provide information on the electronelectron relaxation rate. In defining such parameters f(v) is to be compared with a
Maxwellian that has the same average electron energy, and only the over-all shape of
We thus define a dimensionless velocity variable, z = v/w, in terms
of the characteristic velocity, w, defined by
f(v) is important.
3
-w
4
v 4 f(v, t) dv.
(t) = 4rr
00
Here f(v, t) is normalized so that 4r fO
velocities, g(z, t), is now given by
v2
f dv = 1. The distribution function for scaled
g(z, t) = w 3f(wz, t)
which satifies both 4rr
z 2g dz = 1 and 4-r
zg
dz =
.
This function is to be com-
pared with the correspondingly scaled Maxwellian,
S
g(z) = n
-3/2 e-z
Perhaps the most classic measure of the departure of a velocity distribution from a
Maxwellian is Boltzmann's parameter H. It represents a generalization of the concept
of entropy to systems not in thermal equilibrium. For the scaled distribution function,
H is given by
z g ln g dz
H(t) = 4Tr
0
^
3
which is to be compared with the Maxwellian (minimum) value, H = - (ln rr+1). Other
less physically justifiable parameters may also be used. For example, the quasimoments given by
Qt0
Qn(t) = 4Tr
zZ+n
g dz
z
can (for n# 0, 2) differ from the corresponding Maxwellian values
3+n
Finally, a crude gauge of the departure from Maxwellian may be determined from
the mean absolute difference,
QPR No. 84
138
(XX.
A0 (t) = 4Tr
o
A
0
GASEOUS ELECTRONICS)
zZ Ig-g1 dz.
A
Thus H - H, Qn - Qn, and Ao are all quantities that should approach zero as f(v, t)
Semi-logarithmic plots of the time dependence of these quan-
relaxes to equilibrium.
The open squares represent experimental values based
tities are shown in Fig. XX-1.
H-H
0
Q3 -Q 3
I II II II
SI
Fig. XX- 1.
I
I
I
2
0
4
I
I
i
A0
I
8
6
t (p sec)
10
12
14
Afterglow time dependence of quantities that measure
the departure of f(v) from a Maxwellian.
on the same data (p = 0. 72 Torr, n= 1. 1 X 100 cm-)
discussed in the previous reports.1 ' 2
In Fig. XX-1, the vertical positions of some of the plotted quantities have been shifted
for purposes of display.
In comparing the observed decay rates with a theoretical model, at this stage, we are
still dependent upon solution of a steady-state problem which provides f = a f(v, t) only
at the beginning of the afterglow
(t=0).
Nonetheless,
dependence of the scaled distribution is given by
g= w
3
QPR No. 84
w
3-fw
w
+-vf'+f
W
'
a
3
+W f
e'
139
if f and f are known, the time
(XX.
GASEOUS ELECTRONICS)
where f' is the partial derivative of f(v, t) with respect to v.
separated f into fa'
attributable
to collisions
electron-electron interactions only.
In this expression we have
with atoms, and fe,
attributable
to
In the absence of electron-atom impacts or for a
collision frequency independent of velocity, the term in brackets vanishes.
When H, Qn and Ao are evaluated from g at t = 0, positive theoretical slopes are
obtained.
Initially, electron-atom effects are dominant,
farther away from a Maxwellian.
Indeed,
and the distribution is driven
Experimentally this has never been seen to occur.
simplified analysis of the electron-atom collision-dominated case shows that
the initial trend at t = 0 is quickly reversed in a time of approximately 1
which Maxwellization can occur.
p.sec after
Thus it seems inadvisable to compare the instanta-
neous theoretical rates at t = 0 with the observed relaxation of these parameters.
the other hand, if,
depend,
after some interval, collisions with atoms cease to dominate,
for the most part on fe (v, t).
that
slopes
obtained
g will
The relaxation rates resulting from electron-
electron effects are not expected to vary much with time.
prising
On
It is therefore not too sur-
simply by using g = w fe(t=O) (shown as solid lines in
Fig. XX-1) agree fairly well with the experiment.
We find, then, that the time dependence of parameters that measure the departure
of f(v, t) from a Maxwellian can be correctly given by our simple collisional model,
provided electron-electron interactions dominate.
Reconciling this fact with the theo-
retical result that electron-atom collisions are initially important will require a full
time-dependent solution of the model equation.
B. L.
Wright
References
1.
B. L. Wright, "Microwave Measurements of a Time-dependent Electron Velocity
Distribution Function," Quarterly Progress Report No. 80, Research Laboratory of
Electronics, M.I.T., January 15, 1966, pp. 99-103.
2.
B. L. Wright, "Comparison of Measured Time-dependent Electron Velocity Distributions with a Theoretical Model," Quarterly Progress Report No. 83, Research Laboratory of Electronics, M. I. T., October 15, 1966, pp. 59-64.
QPR No. 84
140